| L(s) = 1 | − 5-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s + 23-s + 25-s − 6·29-s − 8·31-s − 2·37-s + 6·41-s − 4·43-s − 7·49-s − 6·53-s − 4·55-s + 12·59-s − 10·61-s + 2·65-s + 4·67-s − 8·71-s + 10·73-s − 8·79-s + 12·83-s + 2·85-s − 10·89-s − 4·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 0.216·85-s − 1.05·89-s − 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.37085882033992897548010016589, −6.92399045288012367832358368268, −6.12408747621321763573388184921, −5.33623181841178862283214157471, −4.62335081571718190617093864660, −3.80464597157132136070917096216, −3.28180344429350103244922499964, −2.15124254612746542627463086314, −1.26614062143876829013776428956, 0,
1.26614062143876829013776428956, 2.15124254612746542627463086314, 3.28180344429350103244922499964, 3.80464597157132136070917096216, 4.62335081571718190617093864660, 5.33623181841178862283214157471, 6.12408747621321763573388184921, 6.92399045288012367832358368268, 7.37085882033992897548010016589
